Why does one object cut through and show in another object?
I don't see a huge problem, but I can understand that you want to get rid of it. When you have proportional editing enabled,, snapping only has an effect on the selected vertices, not on the ones being dragged along. Also snapping only works if you use g or the widget to move your vertices. If you want your freshly extruded faces to snap to your topology, release them from extrude move mode and use the tools to move them.